TheSverdrup balance, orSverdrup relation, is a theoretical relationship between thewind stress exerted on the surface of the openocean and the vertically integratedmeridional (north-south) transport of ocean water.

Aside from the oscillatory motions associated withtidal flow, there are two primary causes of large scale flow in the ocean:(1)thermohaline processes, which induce motion by introducing changes at the surface intemperature andsalinity, and therefore inseawater density, and(2) wind forcing. In the 1940s, whenHarald Sverdrup was thinking about calculating the gross features of ocean circulation, he chose to consider exclusively the wind stress component of the forcing. As he says in his 1947 paper, in which he presented the Sverdrup relation, this is probably the more important of the two. After making the assumption that frictional dissipation is negligible, Sverdrup obtained the simple result that the meridional mass transport (theSverdrup transport) is proportional to thecurl of the wind stress. This is known as the Sverdrup relation;

${\displaystyle V=\hat{\mathit{z}}\cdot \frac{\mathbf{\nabla }\times \mathit{\tau }}{\beta }}$.

Here,

${\displaystyle \beta }$ is the rate of change of theCoriolis parameter,f, with meridional distance;

${\displaystyle V}$ is the vertically integrated meridionalmass transport including the geostrophic interior mass transport and the Ekman mass transport;

${\displaystyle \hat{\mathit{z}}}$ is theunit vector in the vertical direction;

${\displaystyle \mathit{\tau }}$ is the wind stress vector.

Sverdrup balance may be thought of as a consistency relationship for flow which is dominatedby the Earth's rotation. Such flow will be characterized by weak rates of spin comparedto that of the earth.Any parcel at rest with respect to the surface of the earth must match the spin of the earth underneath it. Looking down on the earth at the north pole, this spin is in a counterclockwise direction, which is defined aspositive rotation or vorticity. At the south pole it is in a clockwise direction, corresponding tonegative rotation. Thus to move a parcel of fluid from the south to the north without causing it to spin, it is necessary to add sufficient (positive)rotation so as to keep it matched with the rotation of the earth underneath it. The left-hand side of the Sverdrup equation represents the motion required to maintain this match between the absolute vorticity of a water column and the planetary vorticity, whilethe right represents the applied force of the wind.

The Sverdrup relation can be derived from the linearizedbarotropic vorticity equation for steady motion:

${\displaystyle \beta v_g=f\mathrm{\partial }w/\mathrm{\partial }z}$.

Here${\displaystyle v_g}$ is the geostrophic interior y-component (northward) and${\displaystyle w}$ is the z-component (upward) of the water velocity. In words, this equation says that as a vertical column of water is squashed, it moves toward the Equator; as it is stretched, it moves toward the pole. Assuming, as did Sverdrup, that there is a level below which motion ceases, the vorticity equation can be integrated from this level to the base of the Ekman surface layer to obtain:

${\displaystyle \beta V_g=f\rho w_E}$,

where${\displaystyle \rho }$ is seawater density,${\displaystyle V_g}$ is the geostrophic meridional mass transport and${\displaystyle w_E}$ is the vertical velocity at the base of theEkman layer.

The driving force behind the vertical velocity${\displaystyle w_E}$ is theEkman transport, which in the Northern (Southern) hemisphere is to the right (left) of the wind stress; thus a stress field with a positive (negative) curl leads to Ekman divergence (convergence), and water must rise from beneath to replace the old Ekman layer water. The expression for thisEkman pumping velocity is

${\displaystyle \rho w_E=\hat{\mathit{z}}\cdot [\mathbf{\nabla }\times (\mathit{\tau }/f)]}$,

which, when combined with the previous equation and adding the Ekman transport, yields the Sverdrup relation.

In 1948Henry Stommel proposed a circulation for the entire ocean depth by starting with the same equations as Sverdrup but adding bottom friction, and showed that the variation inCoriolis parameter with latitude results in a narrowwestern boundary current inocean basins. In 1950,Walter Munk combined the results ofRossby (eddy viscosity), Sverdrup (upper ocean wind driven flow), and Stommel (western boundary current flow), and proposed a complete solution for the ocean circulation.^{[citation needed]}

• Atlantic meridional overturning circulation

• Sverdrup, H.U. (November 1947)."Wind-Driven Currents in a Baroclinic Ocean; with Application to the Equatorial Currents of the Eastern Pacific".Proc. Natl. Acad. Sci. U.S.A.33 (11):318–26.Bibcode:1947PNAS...33..318S.doi:10.1073/pnas.33.11.318.PMC 1079064.PMID 16588757.
• Munk, Walter H. (April 1950)."On the Wind-Driven Ocean Circulation".Journal of the Atmospheric Sciences.7 (2):79–93.doi:10.1175/1520-0469(1950)007<0080:OTWDOC>2.0.CO;2. Retrieved17 February 2025.
• Gill, A.E. (1982).Atmosphere-Ocean Dynamics. Academic Press.ISBN 978-0-12-283522-3.

• Glossary of Physical Oceanography and Related Disciplines Sverdrup balance

• Ocean currents
• Physical oceanography

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